Estimating chaos and complex dynamics in an insect population
TL;DR: In this article, a model- predicted sequence of transitions (bifurcations) in the dynamic behavior of a population from stable equilibria to quasiperiodic and periodic cycles to chaos to three-cycles using cultures of the flour beetle Tribolium.
Abstract: A defining hypothesis of theoretical ecology during the past century has been that population fluctuations might largely be explained by relatively low-dimensional, non- linear ecological interactions, provided such interactions could be correctly identified and modeled. The realization in recent decades that such nonlinear interactions might result in chaos and other exotic dynamic behaviors has been exciting but tantalizing, in that attri- buting the fluctuations of a particular real population to the complex dynamics of a particular mathematical model has proved to be an elusive goal. We experimentally tested a model- predicted sequence of transitions (bifurcations) in the dynamic behavior of a population from stable equilibria to quasiperiodic and periodic cycles to chaos to three-cycles using cultures of the flour beetle Tribolium. The predictions arose from a system of difference equations (the LPA model) describing the nonlinear life-stage interactions, predominantly cannibalism. We built a stochastic version of the model incorporating demographic vari- ability and obtained conditional least-squares estimates for the model parameters. We gen- erated 2000 ''bootstrapped data sets'' with a time-series bootstrap technique, and for each set we reestimated the model parameters. The resulting 2000 bootstrapped parameter vectors were used to obtain confidence intervals for the model parameters and estimated distri- butions of the Liapunov exponents for the deterministic portion (the skeleton) of the model as well as for the full stochastic model. Frequency distributions of estimated dynamic behaviors of the skeleton at each experimental treatment were produced. For one treatment, over 83% of the bootstrapped parameter estimates corresponded to chaotic attractors, and the remainder of the estimates yielded high-period cycles. The low-dimensional skeleton accounted for at least 90% of the variability in the population abundances and accurately described the responses of populations to experimental demographic manipulations, in- cluding treatments for which the predicted dynamic behavior was chaos. Demographic stochasticity described the remaining noise quite well. We conclude that the fluctuations of experimental flour beetle populations are explained largely by known nonlinear forces involving cannibalistic-stage interactions. Claims of dynamic behavior such as periodic cycles or chaos must be accompanied by a consideration of the reliability of the estimated parameters and a realization that the population fluctuations are a blend of deterministic forces and stochastic events.
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01 Jan 2005TL;DR: In this paper, the authors present a spatial analysis of complete point location data, including points, lines, and graphs, and a multiscale analysis of the data set, including spatial diversity analysis and spatial autocorrelation.
Abstract: Preface 1. Spatial concepts and notions 2. Ecological and spatial processes 3. Points, lines and graphs 4. Spatial analysis of complete point location data 5. Contiguous units analysis 6. Spatial analysis of sample data 7. Spatial relationship and multiscale analysis 8. Spatial autocorrelation and inferential tests 9. Spatial partitioning: spatial clusters and boundary detection 10. Spatial diversity analysis 11. Spatio-temporal analysis 12. Closing comments and future directions References Index.
1,250 citations
TL;DR: This work states that because anthropogenic changes often affect stability and diversity simultaneously, diversity-stability relationships cannot be understood outside the context of the environmental drivers affecting both.
Abstract: Understanding the relationship between diversity and stability requires a knowledge of how species interact with each other and how each is affected by the environment. The relationship is also complex, because the concept of stability is multifaceted; different types of stability describing different properties of ecosystems lead to multiple diversity-stability relationships. A growing number of empirical studies demonstrate positive diversity-stability relationships. These studies, however, have emphasized only a few types of stability, and they rarely uncover the mechanisms responsible for stability. Because anthropogenic changes often affect stability and diversity simultaneously, diversity-stability relationships cannot be understood outside the context of the environmental drivers affecting both. This shifts attention away from diversity-stability relationships toward the multiple factors, including diversity, that dictate the stability of ecosystems.
1,247 citations
TL;DR: This work discusses recent advances in understanding ecological dynamics and testing theory using long-term data and reviews how dynamical forces interact to generate some central field and laboratory time series.
Abstract: Both biotic interactions and abiotic random forcing are crucial influences on population dynamics. This frequently leads to roughly equal importance of deterministic and stochastic forces. The resulting tension between noise and determinism makes ecological dynamics unique, with conceptual and methodological challenges distinctive from those in other dynamical systems. The theory for stochastic, nonlinear ecological dynamics has been developed alongside methods to test models. A range of dynamical components has been considered-density dependence, environmental and demographic stochasticity, and climatic forcing-as well as their often complex interactions. We discuss recent advances in understanding ecological dynamics and testing theory using long-term data and review how dynamical forces interact to generate some central field and laboratory time series.
587 citations
TL;DR: The results demonstrate that current estimates of extinction risk for natural populations could be greatly underestimated because variability has been mistakenly attributed to the environment rather than the demographic factors described here that entail much higher extinctionrisk for the same variability level.
Abstract: Extinction risk in natural populations depends on stochastic factors that affect individuals, and is estimated by incorporating such factors into stochastic models. Stochasticity can be divided into four categories, which include the probabilistic nature of birth and death at the level of individuals (demographic stochasticity), variation in population-level birth and death rates among times or locations (environmental stochasticity), the sex of individuals and variation in vital rates among individuals within a population (demographic heterogeneity). Mechanistic stochastic models that include all of these factors have not previously been developed to examine their combined effects on extinction risk. Here we derive a family of stochastic Ricker models using different combinations of all these stochastic factors, and show that extinction risk depends strongly on the combination of factors that contribute to stochasticity. Furthermore, we show that only with the full stochastic model can the relative importance of environmental and demographic variability, and therefore extinction risk, be correctly determined. Using the full model, we find that demographic sources of stochasticity are the prominent cause of variability in a laboratory population of Tribolium castaneum (red flour beetle), whereas using only the standard simpler models would lead to the erroneous conclusion that environmental variability dominates. Our results demonstrate that current estimates of extinction risk for natural populations could be greatly underestimated because variability has been mistakenly attributed to the environment rather than the demographic factors described here that entail much higher extinction risk for the same variability level.
485 citations
TL;DR: In this article, the authors derived three properties of stochastic multispecies communities that measure different characteristics associated with community stability using first-order multivariate autoregressive (MAR(1)) models.
Abstract: Natural ecological communities are continuously buffeted by a varying environment, often making it difficult to measure the stability of communities using concepts requiring the existence of an equilibrium point. Instead of an equilibrium point, the equilibrial state of communities subject to environmental stochasticity is a stationary distribution, which is characterized by means, variances, and other statistical moments. Here, we derive three properties of stochastic multispecies communities that measure different characteristics associated with community stability. These properties can be estimated from multispecies time-series data using first-order multivariate autoregressive (MAR(1)) models. We demonstrate how to estimate the parameters of MAR(1) models and obtain confidence intervals for both parameters and the measures of stability. We also address the problem of estimation when there is observation (measurement) error. To illustrate these methods, we compare the stability of the planktonic commun...
478 citations
Cites methods from "Estimating chaos and complex dynami..."
...The first uses bootstrapping (Efron and Tibshirani 1993, Dennis et al. 2001)....
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References
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31 Jan 1986TL;DR: Numerical Recipes: The Art of Scientific Computing as discussed by the authors is a complete text and reference book on scientific computing with over 100 new routines (now well over 300 in all), plus upgraded versions of many of the original routines, with many new topics presented at the same accessible level.
Abstract: From the Publisher:
This is the revised and greatly expanded Second Edition of the hugely popular Numerical Recipes: The Art of Scientific Computing. The product of a unique collaboration among four leading scientists in academic research and industry, Numerical Recipes is a complete text and reference book on scientific computing. In a self-contained manner it proceeds from mathematical and theoretical considerations to actual practical computer routines. With over 100 new routines (now well over 300 in all), plus upgraded versions of many of the original routines, this book is more than ever the most practical, comprehensive handbook of scientific computing available today. The book retains the informal, easy-to-read style that made the first edition so popular, with many new topics presented at the same accessible level. In addition, some sections of more advanced material have been introduced, set off in small type from the main body of the text. Numerical Recipes is an ideal textbook for scientists and engineers and an indispensable reference for anyone who works in scientific computing. Highlights of the new material include a new chapter on integral equations and inverse methods; multigrid methods for solving partial differential equations; improved random number routines; wavelet transforms; the statistical bootstrap method; a new chapter on "less-numerical" algorithms including compression coding and arbitrary precision arithmetic; band diagonal linear systems; linear algebra on sparse matrices; Cholesky and QR decomposition; calculation of numerical derivatives; Pade approximants, and rational Chebyshev approximation; new special functions; Monte Carlo integration in high-dimensional spaces; globally convergent methods for sets of nonlinear equations; an expanded chapter on fast Fourier methods; spectral analysis on unevenly sampled data; Savitzky-Golay smoothing filters; and two-dimensional Kolmogorov-Smirnoff tests. All this is in addition to material on such basic top
12,662 citations
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01 Jan 1965TL;DR: Algebra of Vectors and Matrices, Probability Theory, Tools and Techniques, and Continuous Probability Models.
Abstract: Algebra of Vectors and Matrices. Probability Theory, Tools and Techniques. Continuous Probability Models. The Theory of Least Squares and Analysis of Variance. Criteria and Methods of Estimation. Large Sample Theory and Methods. Theory of Statistical Inference. Multivariate Analysis. Publications of the Author. Author Index. Subject Index.
8,300 citations
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21 Aug 1973
TL;DR: Preface vii Preface to the Second Edition Biology Edition 1.
Abstract: Preface vii Preface to the Second Edition Biology Edition 1. Intoduction 3 2. Mathematical Models and Stability 13 3. Stability versus Complexity in Multispecies Models 4. Models with Few Species: Limit Cycles and Time Delays 79 5. Randomly Fluctuating Environments 109 6. Niche Overlap and Limiting Similarity 139 7. Speculations 172 Appendices 187 Afterthoughts for the Second Edition 211 Bibliography to Afterthoghts 234 Bibliography 241 Author Index 259 Subject Index 263
5,083 citations
"Estimating chaos and complex dynami..." refers background in this paper
...It was therefore not surprising that the recognition of chaotic dynamics in ecological models (May 1974a, May and Oster 1976) was followed immediately by the search for chaos in existing population time-series data (Hassell et al. 1976)....
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...In particular, two broad classes of stochastic mechanisms important to populations have been widely discussed: demographic stochasticity and environmental stochasticity (May 1974b, Shaffer 1981)....
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...Ecological Monographs Vol. 71, No. 2 of) the largest eigenvalue of Jt evaluated at the equilibrium (the eigenvalue commonly used in stability analysis of a discrete-time system; May 1974b)....
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...For example, the familiar one-dimensional, discrete-time logistic model forecasts dynamical changes (bifurcations) from extinction to equilibrium to two-, four-, eight-cycles, etc., to chaos as the birthrate parameter is increased (May 1974a)....
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